This phrase may have arisen from the practice of leaving an illegitimate baby on the doorstep of the man who was identified as its father.

leave the door open for

ensure that there is still an opportunity for something

open the door to

create an opportunity for

1995 - Kindred Spirit - By recreating the space in which you live or work, Feng Shui can open the door to abundance, wellbeing and a Renewed Sense of Purpose!

show him the door

dismiss someone unceremoniously from your presence

a toe in the door

a first chance of ultimately achieving what you want

a position from which further progress is possible – informal

The image here is of placing your foot in a doorway in such a way as to prevent the door being closed in your face.

dead as a doornail

completely dead

A doornail was one of the large iron studs formerly often used on doors for ornamentation or for added strength. The word occurred in various alliterative phrases (e.g. deaf as a doornail and dour as a doornail) but dead as a doornail is now the only one in common use.

on your doorstep = on the doorstep

very near

close at hand

1998 - New Scientist - The solution to Underbill's problem was on his doorstep.